Image closure of symmetric wide-matrix varieties

Moment Varieties for Mixtures of Products

“…In Section 3 we show that our varieties exhibit finiteness up to symmetry, in the sense of Draisma and collaborators [4,9,10,11]. Namely, if r, d, λ are fixed and n is unbounded then finitely many S n -orbits of polynomials suffice to cut out the varieties σ r (M n,• ).…”

“…Proof. We claim that c (r) is optimal for the linear program (10), with integrality constraints dropped. The dual linear program has variables y S for ∅ = S ⊆ [d] and…”

“…The statement about toric varieties (r = 1) follows from [11,Theorem 1.1]. For r ≥ 2, we apply the main theorem in the recent article [10]. We consider the map (3) between two polynomial rings in countably many variables.…”

“…Solving an integer linear program is expensive in general. However, the integer linear program in (10) has a special structure which allows for a greedy solution that is optimal.…”

“…We conjecture that the integer linear program (10) Informally, the conjecture says that the secant variety has the maximal dimension possible given the sparsity pattern of its parameterization (3). This has been verified in many cases.…”